By Pierre-Antoine Bois; Jean-Pierre Guiraud; et al
The aim of this booklet is to assemble contributions from scientists in fluid mechanics who use asymptotic ways to take care of tricky difficulties. the chosen issues are as follows: vorticity and turbulence, hydrodynamic instability, non-linear waves, aerodynamics and rarefied fuel flows. The final bankruptcy of the publication broadens the viewpoint with an summary of different matters relating asymptotics, offered in a didactic means
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Extra resources for Asymptotic modelling in fluid mechanics : proceedings of a symposium in honour of Professor Jean-Pierre Guiraud, held at the Université Pierre et Marie Curie, Paris France, 20-22 April 1994
89) which is identical with the answer obtained in part (b) of Problem 8 of this chapter, Eq. 81). 10. A particle of mass m is projected upward with a velocity v0 at an angle ˛ to the horizontal in the uniform gravitational field of the earth as shown in Fig. 7. y D 0/ D 0. Fig. 7 Problem 10 y parabola v0 a x (a) Find the Lagrangian in terms of x and y and identify cyclic coordinates . (b) Find the conjugate momenta, identify them and discuss which are conserved and why. 2 Lagrangian and Hamiltonian Dynamics 45 (c) Find the x- and y-components of the velocity as functions of time.
71) Because xR and XR are antiparallel, the net horizontal of m is the sum xR C XR . 73) Putting Eq. 73) into the equation for XR in Eq. 74) Notice that if Â D 0 or if Â D =2, XR vanishes as it should. Moreover, as M ! 1, XR ! 0 as it should. This problem will be worked later in this volume in a tidier fashion using Lagrangian dynamics (see Problem 14, Chap. 2). 10. 75) The motion is restricted to the range =2 < x < 3 =2. (a) Sketch the potential energy function for the region of interest. (b) What is the period T0 of the bounded motion for amplitudes small enough so that the motion can be considered to be simple harmonic?
From Eq. 5) Hamilton’s equations of motion are derived. 6) It is important to remember that to use these formulations the Lagrangian L and the Hamiltonian H must be written in terms of their proper variables. For generalized coordinates qi the Lagrangian must be written in terms of the qi , their time derivatives qP i and possibly the time t. In contrast, the Hamiltonian H must be written in terms of qi and pi . 4) provides the link between pi and qP i . If the Hamiltonian does not explicitly contain the time, then H is a conserved quantity.
Asymptotic modelling in fluid mechanics : proceedings of a symposium in honour of Professor Jean-Pierre Guiraud, held at the Université Pierre et Marie Curie, Paris France, 20-22 April 1994 by Pierre-Antoine Bois; Jean-Pierre Guiraud; et al